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Properties

Label 2-380-95.49-c1-0-8
Degree $2$
Conductor $380$
Sign $0.0314 + 0.999i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.08 + 0.628i)3-s + (−1.99 − 1.01i)5-s − 4.97i·7-s + (−0.710 − 1.23i)9-s − 3.85·11-s + (2.31 − 1.33i)13-s + (−1.53 − 2.35i)15-s + (−2.24 − 1.29i)17-s + (1.24 + 4.17i)19-s + (3.12 − 5.40i)21-s + (1.88 − 1.08i)23-s + (2.94 + 4.04i)25-s − 5.55i·27-s + (1.29 + 2.24i)29-s + 7.76·31-s + ⋯
L(s)  = 1  + (0.628 + 0.362i)3-s + (−0.891 − 0.453i)5-s − 1.87i·7-s + (−0.236 − 0.410i)9-s − 1.16·11-s + (0.641 − 0.370i)13-s + (−0.395 − 0.608i)15-s + (−0.544 − 0.314i)17-s + (0.285 + 0.958i)19-s + (0.681 − 1.18i)21-s + (0.392 − 0.226i)23-s + (0.588 + 0.808i)25-s − 1.06i·27-s + (0.240 + 0.416i)29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0314 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0314 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0314 + 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0314 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840028 - 0.814035i\)
\(L(\frac12)\) \(\approx\) \(0.840028 - 0.814035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.99 + 1.01i)T \)
19 \( 1 + (-1.24 - 4.17i)T \)
good3 \( 1 + (-1.08 - 0.628i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.97iT - 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (-2.31 + 1.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.24 + 1.29i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.88 + 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.29 - 2.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 - 2.75iT - 37T^{2} \)
41 \( 1 + (-3.66 + 6.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.55 - 0.895i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.47 + 0.854i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.89 - 3.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.127 - 0.220i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.66 + 2.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.4 + 6.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.85 + 6.68i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.90 + 2.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.52 + 9.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.04iT - 83T^{2} \)
89 \( 1 + (-4.76 - 8.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.72 + 5.61i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86612747577110634771466371727, −10.38187148811792630933653078893, −9.261140801004617505167777044021, −8.120625236344812786709487832409, −7.73381640316030318101506170333, −6.53400267374010065921784418229, −4.89196096335243573290624984012, −3.96546608996440792731305039983, −3.16101270060164721499454477783, −0.73284230922032999957642282323, 2.39383455621143695642918493478, 2.97912538939304259761158329084, 4.71785310856418765758400548656, 5.81736866508273110635827894950, 6.98383490191943879213262632420, 8.247599241806813559047429930719, 8.410776457519620116084239820554, 9.537203549933600356113281290400, 11.00489118838189509319506405307, 11.45040432773120762675823831364

Graph of the $Z$-function along the critical line